Speech foundation models, exemplified by OpenAI's Whisper, have emerged as leaders in speech understanding thanks to their exceptional accuracy and adaptability. However, their usage largely focuses on processing pre-recorded audio, with the efficient handling of streaming speech still in its infancy. Several core challenges underlie this limitation: (1) These models are trained for long, fixed-length audio inputs (typically 30 seconds). (2) Encoding such inputs involves processing up to 1,500 tokens through numerous transformer layers. (3) Generating outputs requires an irregular and computationally heavy beam search. Consequently, streaming speech processing on edge devices with constrained resources is more demanding than many other AI tasks, including text generation. To address these challenges, we introduce Whisper-T, an innovative framework combining both model and system-level optimizations: (1) Hush words, short learnable audio segments appended to inputs, prevent over-processing and reduce hallucinations in the model. (2) Beam pruning aligns streaming audio buffers over time, leveraging intermediate decoding results to significantly speed up the process. (3) CPU/GPU pipelining dynamically distributes resources between encoding and decoding stages, optimizing performance by adapting to variations in audio input, model characteristics, and hardware. We evaluate Whisper-T on ARM-based platforms with 4-12 CPU cores and 10-30 GPU cores, demonstrating latency reductions of 1.6x-4.7x, achieving per-word delays as low as 0.5 seconds with minimal accuracy loss. Additionally, on a MacBook Air, Whisper-T maintains approximately 1-second latency per word while consuming just 7 Watts of total system power.